Problem: Simplify; express your answer in exponential form. Assume $k\neq 0, a\neq 0$. $\dfrac{{(k^{-1}a^{5})^{-2}}}{{(k^{-2}a^{-2})^{4}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(k^{-1}a^{5})^{-2} = (k^{-1})^{-2}(a^{5})^{-2}}$ On the left, we have ${k^{-1}}$ to the exponent ${-2}$ . Now ${-1 \times -2 = 2}$ , so ${(k^{-1})^{-2} = k^{2}}$ Apply the ideas above to simplify the equation. $\dfrac{{(k^{-1}a^{5})^{-2}}}{{(k^{-2}a^{-2})^{4}}} = \dfrac{{k^{2}a^{-10}}}{{k^{-8}a^{-8}}}$ Break up the equation by variable and simplify. $\dfrac{{k^{2}a^{-10}}}{{k^{-8}a^{-8}}} = \dfrac{{k^{2}}}{{k^{-8}}} \cdot \dfrac{{a^{-10}}}{{a^{-8}}} = k^{{2} - {(-8)}} \cdot a^{{-10} - {(-8)}} = k^{10}a^{-2}$